Steel Canyon Tailor (northwest corner of Steel Canyon)
Independence Port Tailor (eastern side of Independence Port, near the Base Portal)
Founders' Falls Tailor (southeast corner of Founders' Falls, as far south as you can go in The Gaspee neighborhood)
Trina the Body Sculptress in Pocket D (on the wooden loft in the Tiki Bar)

You can edit almost anything about your costume--its colors and patterns, the different clothing pieces used in the costume, and even whether your character has a cape or wings. You can also adjust your character's facial and bodily slider scales. You cannot edit your character's height or gender, however. Changing anything about your costume costs certain amounts of influence--editing just the colors of your costume, for instance, is 3400 influence. (Make sure that you have a good amount of influence--at least 200,000--before you visit the Tailor, so you don't get wiped out of influence accidentally or lose any edits because you can't pay for them.)

Logging out in a Tailor building for at least 24 hours will also give you a Tailor Discount Coupon, which can allay some of the cost of editing your costumes. You can only use one Discount Coupon per session, but you can log out in the Tailor for a few days, then break your costume editing into smaller sessions (like only edit the shirt and use a discount coupon, then only edit the pants or skirt and use another discount coupon, etc.) to help reduce the cost of the whole costume edit.

You can switch between costumes in-game with a simple instantaneous flashover animation. This is triggered using your Chatbox: type /cc and follow it with the number that matches the costume you want to change into. The first costume slot you have corresponds to the number 0, the second slot corresponds to 1, and so on. So, if you want to change into your second costume slot, you would type in the following:

/cc 1

However, if you have bought the Magic and/or Superscience Booster Packs (available from the City of Heroes Buy Now page), you have access to special Costume Change Emotes. To use these emotes, you can type in /cc_e and then follow it with the number of the costume you want to change into, then type "cc" plus the term for the costume change emote you want to use. For example, if you want to do a Wonder Woman-esque spin to change into your third costume slot, you would type in the following:

/cc_e 2 ccspin

My article on CoH editions and packs can be found here, and this page also lists the costume change emotes that are part of some of the booster packs.